Mathematically, if you have two vectors
where
In order to check whether a linear combination can result in a specific vector, we put the vectors as the column of a matrix and we solve the systems of linear equations that we have generated.
In order for a transformation to be linear, it must satisfy 2 conditions.
where
In order to check we can either look at the thing and see that they are not multiples of each other:
or we gotta solve the following equation for
You expand the thing:
Then you group by their power of
If the only solution is
Solve the equation given by:
We may need to re-learn decompositions here.
Solve the system of linear equations given by:
where
An eigenspace is the space related to one eigenvalue.
We have two cases:
In the case of one eigenvector, the basis is simply that eigenvector.
In the case of multiple eigenvectors(they should be linearly independent), the basis is
A matrix is called a diagonal matrix if all of its off-diagonal entries are zero. For example:
where
Every symmetric matrix is diagonalizable.
A matrix
This entire process is called diagonalization, its goal is to find an invertible matrix
where:
Turn the matrix in R.E.F. then get the non-zero rows. Those will be the vectors of our basis.
You either re-use the matrix that we used in the row space, and get the columns that have the pivots(they are surely linearly independent).
Or you can use the same process of the row space on
The rank is just the dimensionality of the column space(the number of vectors in its basis). In this case
The nullity of matrix
We can compute it through the following relation:
The null space of a matrix is the set of vectors that go to 0 after the transformation:
We just need to solve the system of linear equations given by the formula above.
The free variables are
We then group by the parameters and generate these two vectors:
We just pose